具有适当退化性系统拟周期解的存在性

发布时间：2018-04-10 11:24

本文选题：单摆方程 + FHE项　；参考：《湖南师范大学》2016年博士论文

【摘要】：本文,我们主要应用有限维KAM理论证明了一类带拟周期项的单摆方程存在拟周期解、一类带FHE项及拟周期项的van der Pol-Mathieu-Duffing方程存在拟周期解,以及研究了带双曲型退化平衡点的三维非线性拟周期系统的拟周期解存在性.全文共包含四章和一个附录.第一章,主要介绍了单摆方程和van der Pol-Mathieu-Duffing方程的由来和相关的研究背景、实际应用意义,并且介绍了KAM理论的相关背景和国内外研究现状.同时,引出本文的研究内容及意义.第二章,利用有限维KAM理论证明了一类带拟周期项的单摆方程存在拟周期解.首先,我们概述了单摆模型的背景和研究现状,之后我们给出了本文所要研究的一类单摆方程,并且给出了四个KAM定理.我们先证明了一个迭代引理,然后利用此迭代引理来证明主要结论.第三章，，讨论一类带FHE项及拟周期项的van der Pol-Mathieu-Duffing方程拟周期解的存在性.我们先概述了带FHE项及拟周期项的van der Pol-Mathieu-Duffing方程的背景和本文中需要研究的模型,并且给出了一个KAM定理,即在四维相空间中证明了上述方程存在拟周期解.为了在四维相空间中讨论拟周期解的存在性,我们需要做平移、伸缩、极坐标等一系列变换,把方程化成四维相空间中的标准型.然后利用KAM理论来证明此标准型存在拟周期解.从而获得原方程拟周期解的存在性.第四章,我们讨论了带双曲型退化平衡点的三维非线性拟周期系统的拟周期解存在性.对于非退化系统,我们一般把线性部分看成主项来对系统进行约化.但是对于退化系统,由于有零特征根的出现,我们不能直接把线性部分看成主项来对系统进行约化.此时我们可以把某一高次项看成主项,对低次小扰动项整体来进行约化,从而证明系统存在拟周期解.附录中,我们列出了本文使用的一些技术性引理.
[Abstract]:In this paper, we mainly use the finite dimensional KAM theory to prove the existence of quasi periodic solutions for a class of simple pendulum equations with quasi periodic terms, and the existence of quasi periodic solutions for a class of van der Pol-Mathieu-Duffing equations with FHE term and quasi periodic term.The existence of quasi periodic solutions for three dimensional nonlinear quasi periodic systems with hyperbolic degenerate equilibrium is studied.The paper consists of four chapters and an appendix.In the first chapter, the author introduces the origin of simple pendulum equation and van der Pol-Mathieu-Duffing equation, the relevant research background, practical application significance, and introduces the relevant background of KAM theory and the current research situation at home and abroad.At the same time, the research content and significance of this paper are introduced.In chapter 2, the existence of quasi-periodic solutions for a class of simple pendulum equations with quasi-periodic terms is proved by using the finite dimensional KAM theory.First, we summarize the background and research status of the simple pendulum model, then we give a class of simple pendulum equations to be studied in this paper, and give four KAM theorems.We first prove an iterative Lemma and then use this Lemma to prove the main results.In chapter 3, we discuss the existence of quasi periodic solutions for a class of van der Pol-Mathieu-Duffing equations with FHE term and quasi periodic term.First, we summarize the background of van der Pol-Mathieu-Duffing equation with FHE term and quasi periodic term and the model to be studied in this paper. We also give a KAM theorem, that is, the existence of quasi periodic solution of the above equation in four dimensional phase space is proved.In order to discuss the existence of quasi periodic solution in four dimensional phase space, we need to make a series of transformations such as translation, expansion and polar coordinates, and transform the equation into a standard form in four dimensional phase space.Then the existence of quasi periodic solutions of the canonical form is proved by KAM theory.The existence of quasi-periodic solution of the original equation is obtained.In chapter 4, we discuss the existence of quasi periodic solutions for three dimensional nonlinear quasi periodic systems with hyperbolic degenerate equilibrium points.For non-degenerate systems, we generally treat the linear part as the main term to reduce the system.However, for degenerate systems, due to the presence of zero eigenvalues, we can not directly treat the linear part as the main term to reduce the system.In this case, we can treat a higher order term as the main term and reduce the lower order small perturbation term as a whole, thus prove the existence of quasi periodic solutions for the system.In the appendix, we list some technical lemmas used in this paper.
【学位授予单位】：湖南师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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